The topological characterization of band spectra by Chern integers was introduced by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) [1] to explain the quantum Hall effect in a 2D periodic potential. The TKNN integers, giving the Hall conductances, satisfy a Diophantine equation (DE) which was later shown [2] to be a general consequence of magnetic (phase-space) translational invariance and to have several implications [3]. The topological characterization was subsequently extended to quasienergy (QE) band spectra of classically nonintegrable Floquet (time-periodic) systems with phase-space translational invariance [4-6] and a DE was derived also for these systems [5]. More recently, topological properties of Floquet systems have attracted much attention (see, e.g., Refs. [7-11]). Chern integers were associated with the QE bands of nonintegrable double kicked rotors depending periodically on an external parameter [9,10]; an exact quantitative meaning of these integers was given in terms of adiabatic transport in momentum space when the external parameter is slowly varied [9]. In this work, we show that the adiabatically varied double kicked rotors exhibit phase-space translational invariance which allows to derive a DE for the Chern integers, involving both classical and quantum parameters [12].

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[2] I. Dana, Y. Avron, and J. Zak, J. Phys. C 18, L679 (1985); I. Dana and J. Zak, Phys. Rev. B 32, 3612 (1985).

[3] I. Dana, Phys. Rev. B 89, 205111 (2014), and references therein.

[4] P. Leboeuf, J. Kurchan, M. Feingold, and D.P. Arovas, Phys. Rev. Lett. 65, 3076 (1990); Chaos 2, 125 (1992).

[5] I. Dana, Phys. Rev. E 52, 466 (1995).

[6] I. Dana, M. Feingold, and M. Wilkinson, Phys. Rev. Lett. 81, 3124 (1998); I. Dana, Y. Rutman, and M. Feingold, Phys. Rev. E 58, 5655 (1998).

[7] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).

[8] N.H. Lindner, G. Refael, and V. Galitski, Nature Physics 7, 490 (2011).

[9] D.Y.H. Ho and J.B. Gong, Phys. Rev. Lett. 109, 010601 (2012).

[10] H.L. Wang, D.Y.H. Ho, W. Lawton, J. Wang, and J.B. Gong, Phys. Rev. E 88, 052920 (2013).

[11] M. Lababidi, I. Satija, E. Zhao, Phys. Rev. Lett. 112, 026805 (2014); D.Y.H. Ho and J.B. Gong, Phys. Rev. B 90, 195419 (2014).

[12] I. Dana, to be published.